Conditioned stochastic stability of invariant measures on hyperbolic sets
File(s)
Author(s)
Bassols Cornudella, Bernat
Type
Thesis
Abstract
We propose the notion of conditioned stochastic stability to identify invariant measures on repellers that are robust under small random perturbation. This generalises the classical concept of stochastic stability and establishes a rigorous mathematical foundation for the study of chaotic transients in the small noise regime.
Conditioned stochastic stability considers whether quasi-ergodic measures of absorbing Markov processes, generated by small random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure of the deterministic system on the repeller, in the zero noise limit.
Under suitable choices of the random perturbation, we find that equilibrium states from the thermodynamic formalism are conditioned stochastically stable. We establish this result first for uniformly expanding repellers and then for uniformly hyperbolic sets, allowing for the existence of contracting directions.
Conditioned stochastic stability considers whether quasi-ergodic measures of absorbing Markov processes, generated by small random perturbations of the deterministic dynamics and conditioned upon survival in a neighbourhood of a repeller, converge to an invariant measure of the deterministic system on the repeller, in the zero noise limit.
Under suitable choices of the random perturbation, we find that equilibrium states from the thermodynamic formalism are conditioned stochastically stable. We establish this result first for uniformly expanding repellers and then for uniformly hyperbolic sets, allowing for the existence of contracting directions.
Version
Open Access
Date Issued
2025-07-09
Date Awarded
01/10/2025
License URL
Advisor
Lamb, Jeroen
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/S023925/1
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)